Ternary Sums of Squares and Triangular Numbers

TL;DR

This paper characterizes all positive integer triples that make ternary sums of squares and triangular numbers almost universal, resolving a conjecture and completing the classification of such sums.

Contribution

It provides a complete characterization of triples of positive integers for which the ternary sums are almost universal, resolving a conjecture by Kane and Sun.

Findings

Resolved a conjecture by Kane and Sun.

Characterized all triples for almost universal ternary sums.

Completed the classification of mixed sums of squares and triangular numbers.

Abstract

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(\alpha, \beta, \gamma)$ for which the ternary sums $\alpha x^2 +\beta T_y + \gamma T_z$ represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun \cite[Conjecture 1.19(i)]{KS08} and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.